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Theorem frinxp 5218
Description: Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
frinxp (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)

Proof of Theorem frinxp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3630 . . . . . . . . . . 11 (𝑧𝐴 → (𝑥𝑧𝑥𝐴))
2 ssel 3630 . . . . . . . . . . 11 (𝑧𝐴 → (𝑦𝑧𝑦𝐴))
31, 2anim12d 585 . . . . . . . . . 10 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑥𝐴𝑦𝐴)))
4 brinxp 5215 . . . . . . . . . . 11 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
54ancoms 468 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
63, 5syl6 35 . . . . . . . . 9 (𝑧𝐴 → ((𝑥𝑧𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
76impl 649 . . . . . . . 8 (((𝑧𝐴𝑥𝑧) ∧ 𝑦𝑧) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
87notbid 307 . . . . . . 7 (((𝑧𝐴𝑥𝑧) ∧ 𝑦𝑧) → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
98ralbidva 3014 . . . . . 6 ((𝑧𝐴𝑥𝑧) → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
109rexbidva 3078 . . . . 5 (𝑧𝐴 → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1110adantr 480 . . . 4 ((𝑧𝐴𝑧 ≠ ∅) → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1211pm5.74i 260 . . 3 (((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1312albii 1787 . 2 (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
14 df-fr 5102 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
15 df-fr 5102 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1613, 14, 153bitr4i 292 1 (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948   class class class wbr 4685   Fr wfr 5099   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-fr 5102  df-xp 5149
This theorem is referenced by:  weinxp  5220
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